Using the weak solution of Inverse mean curvature flow, we prove the sharp Minkowski-type inequality for outward minimizing hypersurfaces in Schwarzschild space.

We develop foundational theory for the Laplacian flow for closed G2 structures which will be essential for future study. (1). We prove Shi-type derivative estimates for the Riemann curvature tensor Rm and torsion tensor T along the flow, i.e. that a bound on Λ(x,t)=(|∇T(x,t)|^2g(t)+|Rm(x,t)|^2g(t))^{1/2} will imply bounds on all covariant derivatives of Rm and T. (2). We show that Λ(x,t) will blow up at a finite-time singularity, so the flow will exist as long as Λ(x,t) remains bounded. (3). We give a new proof of forward uniqueness and prove backward uniqueness of the flow, and give some applications. (4). We prove a compactness theorem for the flow and use it to strengthen our long time existence result from (2) to show that the flow will exist as long as the velocity of the flow remains bounded. (5). Finally, we study soliton solutions of the Laplacian flow.

We prove that torsion-free G2 structures are (weakly) dynamically stable along the Laplacian flow for closed G2 structures. More precisely, given a torsion-free G2 structure φ on a compact 7-manifold, the Laplacian flow with initial value cohomologous and sufficiently close to φ will converge to a torsion-free G2 structure which is in the orbit of φ under diffeomorphisms isotopic to the identity. We deduce, from fundamental work of Joyce [18], that the Laplacian flow starting at any closed G2 structure with sufficiently small torsion will exist for all time and converge to a torsion-free G2 structure.

In this paper, we prove the Lipschitz regularity of continuous harmonic maps from an finite dimensional Alexandrov space to a compact smooth Riemannian manifold. This solves a conjecture of F. H. Lin in [36]. The proof extends the argument of Huang-Wang [26].

In this paper, we shall discuss the existence, uniqueness and regularity of harmonic maps from an Alexandrov space into a geodesic space with curvature 􏰢 1 in the sense of Alexandrov.